This invention relates in general to a system and method for achieving efficient frequency-doubling of broadband laser light using achromatic phase-matching, and in particular to an efficient achromatic phase-matching optical system and method which has a tunable group velocity dispersion.
Second-harmonic generation (SHG), the generation of light of twice the frequency of available laser light, has been an essential tool of laser research for many years. It is used widely to generate ultraviolet light because such wavelengths are difficult to generate directly from a laser. Indeed, it is used often to generate visible light from a near-infrared laser because it is easier to generate near-infrared laser light than visible light. In general, it is possible to "frequency-double" light from essentially all visible and near-infrared lasers.
A particular type of laser light that is important to frequency-double is ultrashort laser pulses, which offer high temporal resolution and have broad bandwidths and hence have many important applications. Such applications include ultrafast spectroscopy, coherent control of chemical reactions, laser fusion using short ultraviolet pulses as a driver pulse, laser radar (LIDAR), requiring many wavelengths in all spectral ranges, including the ultraviolet, etc. Conventional frequency-doubling using a SHG process has not been particularly efficient or effective for such short pulses.
The use of SHG processes to frequency-double broadband light that is incoherent has also proved to be difficult and inefficient. In general, ultrashort pulses can be considered broadband light whose frequencies are in phase. In contrast, incoherent light can be considered broadband light whose frequencies are randomly phased. These two types of light are difficult to frequency-double due to their respective large bandwidths. As a result of the large bandwidths, efficient methods for frequency-doubling both of these types of light have not been developed. The drawbacks of conventional attempts to handle such broad bandwidths will be understood from the following description of the SHG process and the various methods conventionally used.
The efficiency of a SHG process, .eta., depends on several factors. A first factor is the nonlinear coefficient of a SHG crystal used. This factor depends on internal properties of the crystal and can only be improved by manipulating the composition of the crystal.
Second, .eta. is proportional to the square of the length of the crystal, L. Thus, thick crystals yield much higher efficiency than thin ones.
Third, .eta. depends on the laser intensity and is, typically, directly proportional to the laser intensity. Consequently, continuous-beam lasers, which have relatively low intensity, frequency-double inefficiently while pulsed lasers, which generally achieve higher intensity, frequency-double more efficiently. In general, the shorter the pulse the more efficiently it frequency-doubles. Picosecond (10.sup.-12 to 10.sup.-9 seconds) pulses are known to frequency-double quite efficiently. Values of .eta. in excess of 50% have been reported.
A problem develops, however, when the duration of the pulse is reduced to the femtosecond (10.sup.-15 to 10.sup.-12 seconds) regime. Despite the very high intensity achievable with such very short pulses, very low values of .eta. have conventionally been achieved for such pulses. With femtosecond pulses, it is common to operate at values of .eta. of .apprxeq.1% when generating ultraviolet light.
The low efficiency associated with femtosecond pulses is a result of deviations in phase-matching of the pulse incident on the SHG crystal. In order for frequency-doubling to take place in the crystal, the refractive index of the input laser light (at the "fundamental" wavelength) must equal the refractive index of the frequency-doubled light to be produced. The refractive index of a crystal varies both with the incidence angle and frequency of the input beam. Different incidence angles must be used to obtain maximum efficiency .eta. for different wavelengths. This requirement will be referred to herein as the "phase-matching condition," or "phase-matching" for short. The efficiency .eta. is strongly peaked with respect to angle for a given wavelength and also with respect to wavelength for a given angle. Thus, only a small range of wavelengths near the exact phase-matching wavelength can still yield high efficiency .eta. for a given angle. The range of wavelengths that achieves high-efficiency frequency-doubling for a single angle is called the crystal's "phase-matching bandwidth" for that angle. If the input laser light contains frequencies outside this bandwidth, such frequencies will not produce their corresponding second harmonic (i.e., will not be frequency-doubled). Thus, the efficiency of the overall process is reduced.
When the crystal bandwidth is greater than the input light bandwidth, the above effect can be neglected. However, when the crystal bandwidth is less than the bandwidth of the input light, the SHG efficiency is proportional to the crystal bandwidth, yielding a fourth factor. In this case, the efficiency can be written approximately as: EQU .eta.&lt;&lt;d.sup.2 IL.sup.2 (.DELTA..lambda..sub.cr /.DELTA..lambda..sub.1)
where d is the nonlinear coefficient of the crystal, I is the light intensity, L is the crystal length, .DELTA..lambda..sub.cr is the bandwidth of the crystal, and .DELTA..lambda..sub.1 is the bandwidth of the light.
Known SHG crystals generally have sufficient bandwidth to frequency-double pulses of a few picoseconds or more (which are usually relatively narrowband), but they lack sufficient bandwidth to efficiently frequency-double femtosecond pulses, which are necessarily very broadband, often having bandwidths of many nanometers.
The bandwidth .DELTA..lambda..sub.cr of an SHG crystal is given by: EQU .DELTA..lambda..sub.cr =(.lambda./4l)/[(dn/d.lambda.).sub.f -(dn/d.lambda.).sub.s ]
where .lambda. is the wavelength of light and dn/d.lambda. is the derivative of the refractive index n with respect to wavelength at the appropriate polarization at the fundamental wavelength or second harmonic wavelength, as indicated by the subscripts, f and s, respectively.
Thus, the bandwidth of an SHG crystal is a function of the crystal's refractive-index vs. wavelength curve, a fundamental property of the crystal. Furthermore, another important characteristic of the phase-matching bandwidth, as seen in the above equation, is that it is inversely proportional to the length, L, of the crystal.
As can be appreciated from the above discussion, if a thick crystal is used in order to take advantage of the L.sup.2 dependence, the phase-matching bandwidth will be small due to the 1/L factor in it. In this case, the range of frequencies that fall within the crystal bandwidth will produce second harmonic frequencies efficiently, but the remainder of the pulse frequencies, representing a significant fraction of the pulse's energy, will not produce second harmonic frequencies at all. This yields an overall low value of .eta.. On the other hand, if the crystal is kept thin in order to achieve a large phase-matching bandwidth, then all pulse frequency components will be phase-matched and produce second harmonics, but the L.sup.2 dependence in the efficiency will cause lower values of .eta..
Conventionally, thin-crystals generally produce even lower efficiency than thick-crystals due to the stronger dependence of .eta. on the L (L.sup.2) than phase-matching bandwidth (1/L). However, thin-crystals are often preferred because a second-harmonic pulse with a small range of frequencies (that is, a small bandwidth) is a lengthened pulse, which is generally even more undesirable than an inefficient process. For a 600-nm pulse that is 100 fs in length, the required thickness of KDP (KH.sub.2 PO.sub.4, a commonly used SHG crystal) that does not yield pulse lengthening is about 250 .mu.m. The analogous thickness of BBO (betabarium borate) is approximately 80 .mu.m.
One attempt to solve the above problems is to produce ultra-thin SHG crystals with very high nonlinear coefficients. Typically, organic media are used. However, useful crystals that can compete with standard crystals, such as BBO and KDP have not been achieved.
Various other attempts have been employed such as selecting the refractive-index parameters of KDP to yield a large bandwidth for wavelengths of particular interest, increasing the bandwidth by using two crystals in quadrature, etc. None of these attempts have achieved a satisfactory SHG process.
Another attempt to improve the efficiency of the SHG process is to carry out achromatic phase-matching of the laser pulse incident upon the SHG crystal. FIGS. 1 and 2 show two such conventional approaches. As illustrated, the input light beam 102 is dispersed into its individual frequency components 104 using a diffraction grating 106. As a result, the frequency components 104 of the input light will each propagate at a different angle, with adjacent frequencies having adjacent angles of propagation. Then, using a single lens 108 (FIG. 1) or a two-lens telescope 208 (FIG. 2), these light rays are recombined at the SHG crystal 110. In this manner, all frequencies overlap at the same point and each frequency enters the crystal 110 at its optimal phase-matching angle. Thus, each frequency component of the laser pulse efficiently frequency doubles. In other words, each frequency component essentially acts as an independent and narrowband process, each of which can be quite efficient when a relatively thick crystal is used. Since each frequency component can be treated as a narrowband beam that does not require an SHG crystal with a large bandwidth, a relatively thick crystal can be used.
It is important to note that, because the second harmonic beam produced will be dispersed at an angle, an analogous optical apparatus must be used on the output side of the crystal to reconstruct the second-harmonic beam.
While these designs potentially achieve improved efficiency in the SHG process itself, they introduce a new inefficiency associated with the diffraction grating. Diffraction gratings are not particularly efficient, and since an additional diffraction grating is required to reconstruct the second harmonic on the other side of the SHG crystal, efficiency is reduced even more. This is especially true if the diffraction grating must operate on ultraviolet light, which will be the most common case in SHG processes. When the inefficiencies of the diffraction gratings 106 are considered, the overall efficiency of the SHG process is reduced by roughly a factor of 4. While the overall efficiency of these designs is still greater than that typically obtainable without achromatic phase-matching using standard crystals, the efficiency is not sufficiently improved that achromatic phase-matching has found practical use.
An alternative approach uses prisms instead of diffraction gratings to disperse the input beam. Both disperse light into its frequency components, but prisms can be anti-reflection-coated or used at Brewster's angle and hence, can result in no loss of efficiency. However, prisms typically have about one tenth the dispersion available from a grating, which is required for a typical achromatic phase-matching situation.
As illustrated in FIGS. 3 and 4, in such designs the prisms have been used in conjunction with lens devices to amplify the prism dispersion to appropriate values. In FIG. 3, the input light 301 is incident on a single prism 303 and a two-lens telescope 305 is used to amplify the dispersion of the prism 303 and to focus the light onto the SHG crystal 307. In FIG. 4, the light pulse 401 is passed through two oppositely oriented prisms 403, 404 and then directed through a single lens 407 to recombine the various frequencies in the SHG crystal 409.
The device depicted in FIG. 3 achieves sufficient dispersion because the two-lens telescope 305 amplifies the dispersion of the prism by 1/M, where M is the magnification of the telescope. A problem associated with such a design is that the group velocity dispersion (the tendency for red wavelengths to travel faster than blue wavelengths) in the system is always positive. Thus, the pulse spreads in time greatly reducing the efficiency of the overall systems for most types of ultrashort light pulses as more fully described below.
The device of FIG. 4 achieves sufficient dispersion because a sufficiently short-focal-length lens 407 can be used to recombine the spatially dispersed rays out of the two-prism assembly to achieve the desired dispersion. While this design can achieve zero (or negative) group-velocity dispersion, it suffers from a different flaw. The angle at which the light rays are incident at the crystal is dependent upon the input position of the input light beam 401. This also tends to reduce the efficiency of the SHG process as more fully developed below.